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Properties

Label 2-550-11.3-c1-0-12
Degree $2$
Conductor $550$
Sign $0.350 + 0.936i$
Analytic cond. $4.39177$
Root an. cond. $2.09565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.118 + 0.363i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.224i)6-s + (0.927 − 2.85i)7-s + (0.309 + 0.951i)8-s + (2.30 − 1.67i)9-s + (−1.23 − 3.07i)11-s + 0.381·12-s + (−5.04 + 3.66i)13-s + (0.927 + 2.85i)14-s + (−0.809 − 0.587i)16-s + (−3.54 − 2.57i)17-s + (−0.881 + 2.71i)18-s + (−1.80 − 5.56i)19-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.0681 + 0.209i)3-s + (0.154 − 0.475i)4-s + (−0.126 − 0.0916i)6-s + (0.350 − 1.07i)7-s + (0.109 + 0.336i)8-s + (0.769 − 0.559i)9-s + (−0.372 − 0.927i)11-s + 0.110·12-s + (−1.39 + 1.01i)13-s + (0.247 + 0.762i)14-s + (−0.202 − 0.146i)16-s + (−0.859 − 0.624i)17-s + (−0.207 + 0.639i)18-s + (−0.415 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.350 + 0.936i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735043 - 0.509733i\)
\(L(\frac12)\) \(\approx\) \(0.735043 - 0.509733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
11 \( 1 + (1.23 + 3.07i)T \)
good3 \( 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.927 + 2.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.54 + 2.57i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.85T + 23T^{2} \)
29 \( 1 + (0.163 - 0.502i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.26 + 10.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.14 - 3.52i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + (0.454 + 1.40i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.35 - 3.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.07 - 6.37i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.04 + 5.11i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.0901T + 67T^{2} \)
71 \( 1 + (-10.7 - 7.83i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.57 + 10.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.28 + 4.56i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.04 - 2.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (-0.763 + 0.555i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64931511159721196670192245220, −9.465426773853650858798246266838, −9.133163771003091378723852564690, −7.74891599804994822511688363903, −7.15560489844858141095753169786, −6.35134160901839102336078310330, −4.83197046878079002583862819252, −4.15908345950678906327964764053, −2.41060819445890494308896934530, −0.60222613166739171985762927008, 1.86499911203553133826396252197, 2.59941145187097577638555932390, 4.34435503366885156396654507971, 5.29239689561598637077463624090, 6.57992962651494482389499689621, 7.81734780706374642276326921818, 8.072654887682857637070886038868, 9.350384370002206525449296772879, 10.12859744890262876612136299829, 10.69782391937247636616012241501

Graph of the $Z$-function along the critical line